Abstract
It is known that many of the problems of mathematical physics, reduced to a differential equation with partial derivatives written in cylindrical and spherical coordinates, by using method of separation of variables, in particular, leads to the Bessel differential equation and Bessel functions. In practice, especially in problems of electrodynamics, celestial mechanics and modern applied mathematics most commonly used Fourier series in orthogonal systems of special functions. Given this, it is required to determine the conditions of expansion of functions in series into these special functions, forming in a given interval a complete orthogonal system. The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space L 2 := L 2 ([0 , 1] , x dx ) of square summable functions f : [0 , 1] → R with the weight x . The exact inequalities of Jackson–Stechkin type on the sets of L2 ( r ) 2 ( D ) , linking En − 1 ( f ) 2 — the best approximation of function f by partial sums of order n − 1 of the Fourier–Bessel series with the averaged positive weight of generalized modulus of continuity of m order Ω m (
Highlights
The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space L2 := L2([0, 1], x dx) of square summable functions f : [0, 1] → R with the weight x
Similar inequalities are obtained through the K-functionals r-s derivatives of functions
The exact value of the different n-widths for classes of functions defined by specified characteristics, in L2 were calculated
Summary
{Jν (λkx)}∞ k=1 является полной и ортогональной в пространстве суммируемых с квадратом функций f с весом x на отрезке [0,1]. √ откуда вытекает, что система функций {︀ 2Jν(λkx) · |Jν′ (λk)|−1}︀ образует полную ортонормированную систему в пространстве L2. Обозначения, через {Jν(λkx)}∞ k=1 обозначим полную ортонормированную систему функций в пространстве L2, для которой. Для произвольной функции f ∈ L2 рассмотрим разложение в ряд Фурье–Бесселя следующего вида:. Рассмотрим дифференциальный оператор Бесселя второго порядка первого рода индекса ν: d2 1 d ν2 D = dx2 + x · dx − x2 ,. Обозначим через L2(D), где оператор D определяется равенством (3), множество функций f ∈ L2, имеющих абсолютно непрерывные производные первого порядка f ′, и таких, что. Обозначим множество функций f ∈ L2, имеющих абсолютно непрерывные производные (2r − 1)-го порядка и для которых Drf ∈ L2.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
